Albion remain out of the Premier League drop zone tonight.
They are still 17th after Fulham lost 3-0 at home to Manchester City.
But Burnley appear to be easing towards safety after winning at Everton.
Albion will be keen to take care of their own business when they visit Southampton tomorrow.
A point or three for Fulham would have seen the Seagulls into the relegation places for the first time.
They looked potentially on course after a strong first-half display.
But John Stones was left unmarked to turn home a Joao Cancelo free-kick just 75 seconds into the second period.
Gabriel Jesus added the second after more poor defending and Sergio Aguero scored from the penalty spot.
ALBION INVESTIGATE CONNOLLY LOCKDOWN CLAIM
Burnley earlier moved seven points clear of both Fulham and Albion by winning 2-1 at Everton.
They emulated wins by Fulham and Newcastle at Goodison Park in recent weeks.
Dwight McNeill’s brilliant shot into the top corner proved to be the winner.
Chris Wood scored the first for the Clarets.
Dominic Calvert-Lewin headed home to halve arrears for the Toffees before half-time.
But Burnley went closest to scoring in the second period as they saw the job through impressively.
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